(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(b(c(x1)))) → b(b(a(a(x1))))
b(x1) → c(c(a(a(x1))))
b(c(x1)) → a(x1)
a(a(c(x1))) → x1
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(c(z0)))) → b(b(a(a(z0))))
a(a(c(z0))) → z0
b(z0) → c(c(a(a(z0))))
b(c(z0)) → a(z0)
Tuples:
A(a(b(c(z0)))) → c1(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
B(z0) → c3(A(a(z0)), A(z0))
B(c(z0)) → c4(A(z0))
S tuples:
A(a(b(c(z0)))) → c1(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
B(z0) → c3(A(a(z0)), A(z0))
B(c(z0)) → c4(A(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c3, c4
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(b(c(z0)))) → c1(B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(c(z0)))) → b(b(a(a(z0))))
a(a(c(z0))) → z0
b(z0) → c(c(a(a(z0))))
b(c(z0)) → a(z0)
Tuples:
B(z0) → c3(A(a(z0)), A(z0))
B(c(z0)) → c4(A(z0))
S tuples:
B(z0) → c3(A(a(z0)), A(z0))
B(c(z0)) → c4(A(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
B
Compound Symbols:
c3, c4
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 2 of 2 dangling nodes:
B(z0) → c3(A(a(z0)), A(z0))
B(c(z0)) → c4(A(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(b(c(z0)))) → b(b(a(a(z0))))
a(a(c(z0))) → z0
b(z0) → c(c(a(a(z0))))
b(c(z0)) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))